Homological spectral sequence

A homological spectral sequence in an abelian category (Abelian category), starting at a specified integer $r_0\ge 0$, consists of objects $E_{p,q}^r$, maps $d_{p,q}^r:E_{p,q}^r\to E_{p-r,q+r-1}^r,d^r{d}^r=0,$ for r≥r₀ and p,q∈ℤ, and specified isomorphisms ${\alpha }_r:\frac{\ker (d_{p,q}^r)}{\mathrm{im}(d_{p+r,q-r+1}^r)}\stackrel{\sim }{\to }{E}_{p,q}^{r+1}\mathrm{.}$ The quotient has the meaning of Homology object of a chain complex; no direct sum of the whole family is assumed. The isomorphisms are data. First quadrant means the initial page vanishes unless p,q≥0; third quadrant means it vanishes unless p,q≤0. Bounded means that for each integer n only finitely many initial-page terms with p+q=n are nonzero. These support conditions persist because later terms are subquotients of earlier ones.

Weibel, Chapter 5, §5.2 pp.122–125 and Definitions 5.4.2–4 pp.132–133; Sharifi, §4.1 pp.87–89 and Definition 4.2.1 p.90.